Exponential stability of perturbed linear discrete systems

Diblík et al. Advances in Difference Equations (2016) 2016:2 DOI 10.1186/s13662-015-0738-6 RESEARCH Open Access Exponential stability of perturbed linear discrete systems Josef Diblík1*, Denys Y Khusainov2, Jaromír Baštinec1and Andrii S Sirenko2 *Correspondence: [email protected].cz 1Brno University of Technology, Brno, Czech Republic Full list of author information is available at the end of the article Abstract The paper considers the problem of exponential stability and convergence rate to solutions of perturbed linear discrete homogeneous systems. New criteria on exponential stability are derived by using the second method of Lyapunov. We consider non-delayed systems as well as systems with a single delay. Simultaneously, explicit exponential estimates of the solutions are derived. The results are illustrated by examples. 1 Introduction Thedynamics of discontinuous transition statesofdynamic systems is mostnaturallydescribed by difference equations and investigating the properties of difference equations is averyimportantareaofresearch(wementionatleastthemonographs[–]).Ifprocesses are modeled by systems of equations, in general, some of the model parameters may be uncertain and we have to deal with systems with inaccurately specified parameters. Inthepaper,weconsiderthe exponentialstabilityofsystemsofperturbedlinearhomogeneous difference equations and give explicit exponential estimates of solutions. In such systems, itis usuallyappropriatetodescribe theinaccuratelyspecified parameters asperturbationsofsomeinitialsystems.Investigatedarenon-delayedhomogeneoussystemsof differenceequationsandhomogeneoussystemsofdifferenceequationswithasingledelay. LetAbeasymmetricpositivedefinitematrix.Denotebyλmax(A),λmin(A)itsmaximum and minimum eigenvalues and let ϕ(A):=λmax(A) λmin(A). Throughout the paper, for an arbitrary matrix B,weusethematrixnorm |B|=λmaxBTB. This norm reduces to the Euclidean norm |x|=   n  i= x i if B=x=(x,...,xn)T. ©2016 Diblík et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Diblík et al. Advances in Difference Equations (2016) 2016:2 Page 2 of 20 Now we describe the problems solved in the paper and the methods frequently used to investigate stability. 1.1 Non-delayed systems Thesimplestclassofsystemsofdifferenceequationsisthatoflinearautonomoussystems x(k+)=Ax(k), k=,,..., () where A=(aij)n i,j= is a real constant matrix and xis an n-dimensional unknown vector. In the paper, we consider systems x(k+)=A+B(k)x(k), k=,,..., () wherethenorm|B(k)|oftherealmatrixB(k)=(bij(k))n i,j= issmallinacertainsensespecifiedbelow.System()canbeviewedasaperturbationofsystem().Recallthatthesolution of ()or() is uniquely determined by the given initial value x(). Definition  The zero solution x(k)=,k=,,...of ()iscalled(Lyapunov)stableif, for an arbitrary ε>,thereexistsaδ(ε)>suchthat,foranysolutionx∗(k)ofsystem(), we have |x∗(k)|<εfor k=,,...,providedthat|x∗()|<δ(ε). If,moreover, lim k→+∞x∗(k)=, the zero solution is called asymptotically stable. The zero solution of () is called exponentially stable if there existconstants N>andθ∈(,) not depending on x∗such that x∗(k)≤Nx∗()θk,k=,,.... () For the basic notions, properties, and results we refer, e.g.,to[, , , ] and to the references therein. For linear systems, all solutions are simultaneously stable (exponentially stable)ifthezerosolutionisstable(exponentiallystable).Therefore,incontrasttononlinearsystems,forlinearsystems,theconceptofastablesystem(exponentiallystablesystem) is meaningful. The stability of system () is often analyzed through the characteristic equation det(λI–A)=, () where Iis an n×nidentity matrix and λis a suitable complex constant. Expanding (), we obtain λn+pλn– +pλn– +···+pn–λ+pn= () withrealpi,i=,,...,n.Necessaryandsufficientfortheexponentialstabilityofthelinear system () is the condition ρ(A)<(see,e.g.,[, ]) where ρ(A)isthespectralradiusof the matrix A. For discrete equations, unfortunately, such a simple and effective tool of investigation of the roots of the characteristic equation for the differential systems, as the Diblík et al. Advances in Difference Equations (2016) 2016:2 Page 3 of 20 Hurwitz criterion giving the necessary and sufficient condition for Reλi<,i=,,...,n, is not available. The well-known Schur-Cohn criterion [, , ], to verify ρ(A)<,isnot easily applicable because the higher the order of the characteristic equation (), the more difficult the computation becomes. Analternativetostudying thestability by algebraicmethods isthemethod of Lyapunov functions (described, e.g.,in[, ]).Forlinearstationarysystems(), the Lyapunov function is sought in the form of a quadratic form V(x)=xTHx where a symmetric positive definite n×nmatrix His the solution of the matrix Lyapunov equation ATHA–H=–C.() System()isexponentiallystableifandonlyif,foranarbitrarypositivedefinitesymmetric n×nmatrix C, the matrix equation () is solvable and has a unique solution - a positive definite symmetric matrix H[]. The parameters of mathematical models of dynamic systems are, generally speaking, determined experimentallyandareknownwithsomedegree ofadequacy.Thestabilityof such systems can be investigated as the stability of a perturbed system to a given initial system. Theinvestigationofthestabilityofsystem()is, in essence,closetotheinvestigationof what is called interval stability of linear differential equations. Such an investigation was carried out by Haritonov, we refer at least to one of his founding papers []wherethe necessary and sufficient conditions for stability were formulated. 1.2 Systems with a single delay In Section , we study linear homogeneous differencesystems with a single delay, x(k+)=A+B(k)x(k)+D+E(k)x(k–m), k=,,..., () where A=(aij)n i,j= and D=(dij)n i,j= are real constant matrices, m∈N, and norms |B(k)|, |E(k)|of the real matrices B(k)=(bij(k))n i,j=,E(k)=(eij(k))n i,j= are small in a sense. System () can be viewed as a perturbation of the system x(k+)=Ax(k)+Dx(k–m), k=,,.... () Recall that the initial problem x(k)=x∗ k,k=–m,–m+,...,for()or()wherex∗ k∈Rn is uniquely solvable. Below, we utilize the norm x()m:=maxx(i),i=–m,–m+,...,. Definition  The zero solution x(k)=,k=–m,–m+,... of system () is called (Lyapunov) stable if, for an arbitrary ε>,thereexistsaδ(ε)>suchthat,foranysolution x∗(k)ofsystem(), we have |x∗(k)|<εfor k=,,...,providedthatx∗()m<δ(ε). If, moreover, lim k→+∞x∗(k)=, Diblík et al. Advances in Difference Equations (2016) 2016:2 Page 4 of 20 the zero solution is called asymptotically stable. The zero solution of system () is called exponentiallystable if there exist constants N>andθ∈(,) not depending on x∗such that x∗(k)≤Nx∗()mθk,k=,,.... () To establishthe factof the asymptoticstabilityof system(), weconsidertheassociated characteristic equation detλm+I–λmA–D=. Expanding the determinant on the right-hand side, we obtain, in general, an algebraic equation of degree (m+)nand we are facing an even greater difficulty than in the case of equation (). Recently, for the asymptotic stability of special classes of linear discrete equations with constant complex coefficients and with delay, simple criterions have been derived in [–] (the criteria derived in [] are fully explicit with respect to the delay). Thepaperisorganizedasfollows.Sufficientconditionsfortheexponentialstabilityand explicit exponential estimates of solutions of non-delayed systems ()arederivedinSection  by the second method of Lyapunov. For systems with a single delay (), such an investigationisperformedinSection.Theinvestigationsofbothclassesofsystems()and () differ. The difference is caused by the presence of delayin (). The difficulty caused by theLyapunovfunctionbeingestimatedwasovercomebyusingwhatiscalledRazumikhin condition while applying of several auxiliary inequalities. The applicability of the results is illustrated by examples and concluding remarks are included in Section . 2 Exponential stability of non-delayed systems Theorem belowgivessufficientconditionsforexponentialstabilityoftheperturbedsystem(),andsuitableparametersNandθinthedefinitionofexponentialstability()ofsolutions are explicitly defined. In the proof, Lyapunov function V(x)=xTHx is used where Hsolves the matrix Lyapunov equation (). With respect to the matrix of perturbation B(k), k=,,...weassumethatthereexistsanumberDsuch that, for every k=,,..., we have B(k)≤D. First, we need auxiliary lemmas. Denote D∗:=  λmax(H)ATH+λmin(C)λmax(H)–ATH and ξ:=λmin(C)–ATHD–|H|D. Lemma  Let ρ(A)<.Let C be a fixed positive definite matrix and let matrix H solve matrix equation (). Then D∗>and,if D<D∗,then ξ>. Diblík et al. Advances in Difference Equations (2016) 2016:2 Page 5 of 20 Proof Inequality D∗> is obvious. Solve the quadratic equation ξ=withrespecttoD. Its roots D, are D, = λmax(H)–ATH±ATH+λmin(C)λmax(H) and ξ>ifD∈(D,D)=(D,D∗), D< . Since, by its definition, D≥, we conclude that ξ>ifD∈[,D∗).  The following result is taken from Chapter , Section  in []. Lemma Let Aand Abe positive definite matrices.Then λmin(A+A)≥λmin(A)+λmin(A). () IntheRussiantranslationof[],itisnotedthat()holdsevenforarbitraryHermitian matrices A,A. Theorem  Let ρ(A)<.Let C be a fixed positive definite matrix and let matrix H solve the matrix Lyapunov equation (). If D<D∗,() then system () is exponentially stable and,for any of its solutions x=x(k),the inequality x(k)≤– ξ λmax(H)k/ϕ(H)x(),k≥, () holds where <– ξ λmax(H)<. () Proof Recall that His a symmetric positive definite matrix and, by Lemma , ξ>since () is assumed. Define a Lyapunov function V(x):=xTHx. For the first difference of V(x) along the trajectories of system (), we have (k≥) Vx(k)=Vx(k+)–Vx(k) =Ax(k)+B(k)x(k)THAx(k)+B(k)x(k)–xT(k)Hx(k). We transform the last difference to Vx(k)=xT(k)ATHA–Hx(k)+xT(k)ATHB(k)x(k) +xT(k)B(k)THB(k)x(k) Diblík et al. Advances in Difference Equations (2016) 2016:2 Page 6 of 20 ≤–λmin(C)x(k)+ATHDx(k)+|H|Dx(k) =–ξx(k).() Consequently, V(x(k))≤, and the zero equilibrium (and, therefore, system ()) is stable. Now we derive an explicit estimate () of the exponential type. Since λmin(H)|x|≤V(x)≤λmax(H)|x|, () from inequality ()weget Vx(k+)–Vx(k)≤–ξx(k)≤–ξ λmax(H)Vx(k) or Vx(k+)≤– ξ λmax(H)Vx(k).() Below we solve this difference inequality, but first we show that the coefficient – ξ λmax(H) ontheright-handsideof ()satisfiesestimates().Theright-handsideofinequality() is a simple consequence of inequality ξ>. To prove the left-hand side of inequality (), we use Lemma with matrices A,A defined as A:= H–ATHAand A:=ATHA.Then λmin(H)=λminH–ATHA+ATHA =λmin(A+A)≥λmin(A)+λmin(A) =λminH–ATHA+λminATHA and λmin(C)=λminH–ATHA≤λmin(H)–λminATHA ≤λmax(H)–λminATHA<λmax(H). Therefore, ξ=λmin(C)–ATHD–|H|D<λmax(H) and the left-hand side of inequality ()holds. Solving the difference inequality (), we obtain Vx(k)≤– ξ λmax(H)kVx(),k≥. Diblík et al. Advances in Difference Equations (2016) 2016:2 Page 7 of 20 Using () together with the obtained inequality, we get λmin(H)x(k)≤Vx(k)≤– ξ λmax(H)kVx() ≤λmax(H)x()– ξ λmax(H)k,k≥, and, consequently, x(k)≤– ξ λmax(H)kλmax(H) λmin(H)x()=– ξ λmax(H)k ϕ(H)x(),k≥ or x(k)≤– ξ λmax(H)k/ϕ(H)x(),k≥, i.e.,wegetinequality().  Example Let n=and A=. . . . be specified in system (). Let C=. –. –. . . The matrix H=   solves the matrix Lyapunov equation (), λmin(C)=.,λmin(H)=λmax(H)=ϕ(H)=, |H|=,|ATH|=.andD∗=..If|B(k)|≤D<D∗=.,k=,,...,thensystem()is exponentially stable. For, say, D=.,wehaveξ=.and()becomes x(k)≤– ξ λmax(H)k/ϕ(H)x()=(.)k/ ·x(),k≥. 3 Exponential stability and estimation of convergence of linear systems with delay Inthispartwestudylineardifferencesystemswithsingledelay()bythesecondLyapunov method. A short survey of the use of the second Lyapunov method for the investigation of stability of discrete systems with delay now follows. Often, two modifications are used. The first is what is called the method of finite-dimensional Lyapunov functions, and the second one is the method of functionals of Lyapunov-Krasovskii. Strictly speaking, the Diblík et al. Advances in Difference Equations (2016) 2016:2 Page 8 of 20 lattermethodiscalledthemethodoffunctionalsonlybecauseitisadiscreteanalogofthe methodoffunctionalsforfunctionaldifferentialequationswithdelay.Inthefollowing,we use the former method. IfthemethodofLyapunovfunctionsistobeappliedtosystemsofdifferentialequations withdelay ˙ x(t)=f(x(t),x(t–τ)) wheref:Rn×Rn→Rnandτ>,itisnecessarytofinda positivedefiniteandcontinuouslydifferentiablefunctionV(x)suchthatitstotalderivative along solutions of the system is negative definite provided that a so-called Razumikhin condition holds, i.e., that the solutions approach the surface level ∂Vα=x∈Rn:V(x)=α, where αis a positive constant from the domain Vα=x∈Rn:V(x)<α. Formally,theRazumikhinconditioncanbewrittenasV(x(s))< V(x(t))wheret–τ≤s<t. For quadratic Lyapunov functions V(x)=xTHx with a symmetric positive definite matrix H, this inequality implies |x(s)|<ϕ(H)|x(t)|where t–τ≤s<t. Below, we consider a similar approach for studying the stability of difference systems with delay. 3.1 Auxiliary inequalities First we derive three lemmas related to auxiliary inequalities. Such inequalities are used in the proof of Theorem  in Section .. Lemma  Let L>,a≥, b≥, c>,d≥, and e>be constants.If there exist positive constants ξand ηsuch that B(k)<ξ,E(k)<η,k≥, () and ξ<ξ,η<ηwhere ξ=–a+a +cL c,() η=–(b+dξ)+(b+dξ)+eL e,() then L–aB(k)–bE(k)–cB(k)–dB(k)E(k)–eE(k)>, k≥. () Proof If L–aξ–bη–cξ–dξη–eη>, then ()holds.Letξbe such that cξ+aξ– L<. Diblík et al. Advances in Difference Equations (2016) 2016:2 Page 9 of 20 The above inequality holds if  < ξ<ξwhere ξis defined by (). Now we find ηsuch that eη+(b+dξ)η– L<. The last inequality is valid if  <η<ηwhere ηis defined by (). Therefore, ()willbe valid for  <ξ<ξ,<η<ηand k≥.  Lemma  Let L>,a≥, b≥, c>,and d>be constants.If there exist positive constants ξand ηsuch that B(k)<ξ,E(k)<η,k≥, () and ξ<ξ,η<ηwhere ξ=–a+a +cL c,() η=L (b+dξ),() then L–aB(k)–bE(k)–cB(k)–dB(k)E(k)>, k≥. () Proof If L–aξ–bη–cξ–dξη>, then ()holds.Assumethatξsatisfies cξ+aξ– L<. This inequality holds if  <ξ<ξwhere ξis defined by (). Now we find ηsatisfying (b+dξ)η– L<. Thisinequalityisvalidif<η<ηwhereηisdefinedby().Therefore,()willbevalid for <ξ<ξ,<η<η,andk≥.  Lemma Let L>,a≥, b≥, d>,and e>be constants.If there exist positive constants ξand ηsuch that B(k)<ξ,E(k)<η,k≥, () and ξ<ξ,η<ηwhere η=–b+b +eL e,() Diblík et al. Advances in Difference Equations (2016) 2016:2 Page 16 of 20 –|HA|+|HD|+|HA|γmϕ(H)E(k)–|H|B(k) –|H|+γmϕ(H)B(k)E(k)–|H|γmϕ(H)E(k)>. () As follows from (), (), and (), inequality ()canberewrittenas L–aB(k)–bE(k)–cB(k)–dB(k)E(k)–eE(k)>. We apply Lemma with L,a,b,c,d,andedefined by ()-() (obviously c>, e>). It is sufficient to satisfy inequality (), i.e., B(k)<ξ,wherewesetξ=D∗ , E(k)<η,wherewesetη=D∗ . Both inequalities are valid due to assumptions (), (). Consequently, () holds and, from (), we get Vx(k),k<Vx(k+),k+<Vx(k),k. Therefore, assumption () is false and ()holdsforeveryk≥–m.From()and() we get γkλmin(H)|x|≤V(x,k)≤ελmin(H)=δ(ε)λmax(H)=x() mλmax(H), () where k≥,and x(k)≤ϕ(H)x()mγ–k/,k≥, () i.e.,equation()holds.  Example Let n=,m=, A=. . . . and, for simplicity, Dbe a ×nullmatrixspecifiedinsystem(). Let C=. –. –. . . The matrix H=   solvesthematrixLyapunovequation().AsinExample,wegetλmin(C)=.,λmin(H)= λmax(H)=ϕ(H)=,|H|=and|ATH|=..Moreover,letγ=..Then L=λmin(C)–(–/γ)|H|. =., Diblík et al. Advances in Difference Equations (2016) 2016:2 Page 17 of 20 a=|HA|=., b=|HA|+|HA|γϕ(H)=., c=|H|=, d=+γϕ(H)|H|=., e=γ|H|ϕ(H)=., L =L. =., a =a=., b =|HA|=., c =d =|H|=, L =λmax(H)–γλ min(C)–(γ–)|H|. =., a =γ|HA|=., b =γ|HA|=., c =d =γ|H|=., L=λmin(H)=, a=, b=γm+|HA|ϕ(H). =., d=e=γm+|H|ϕ(H). =., and ξ. =., η. =., ξa. =., ηa. =., ξb. =., ηa. =., ξ. =., η. =.. Let <D∗ <D=min{ξ,ξa,ξb,ξ}=ξ. =. and <D∗ <D=min{η,ηa,ηb,η}=η. =.. If |B(k)|≤D∗ ,|E(k)|≤D∗ ,k=,,...,thensystem() is exponentially stable. From (), we get x(k)≤ϕ(H)x()mγ–k/ =(.)–k/ ·x(),k≥. 4 Concluding remarks The paper investigates the exponential stability assuming, without loss of generality, that the initial point is always defined for the value of the independent variable kbeing equal to  in the case of the non-delayed equation (), and that the initial functions in the case of the delayed systems ()arealwaysdefinedfork=–m,–m+,...,.Obviously,our investigation can easily be modified for an arbitrary integer value of independent variable k=kin the case of the non-delayed equation () or if the initial functions, in the case of the delayed systems () are defined for the values of k=k–m,k–m+,...,k.Tothis end, some minor changes in Definitions and arenecessary.Weomitthedetailsaswell as the relevant reformulation of the results. Carefully tracing the proof of Theorem we note that system () will be exponentially stable if condition () is replaced by a more general assumption: limsup k→∞ B(k)<D∗.() Diblík et al. Advances in Difference Equations (2016) 2016:2 Page 18 of 20 Nevertheless, in such a case, inequality () is, in general, not preserved because the constant N:= ϕ(H)in() (see Definition ) can be different. Similarly, the exponential stabilityofsystem()willbepreservedifinequalities(),()inTheoremarereplaced by limsup k→∞ B(k)<D∗ ,() limsup k→∞ E(k)<D∗ .() In such a case, inequality () will be valid if, in general, the constant N:= ϕ(H)in() (see Definition ) is different. An open problem arises if we discuss inequalities ()and (), () concerning their further improvement. Particularly, is it possible to replace () by the weaker condition limsup k→∞ ρB(k)<D∗, and (), () by the weaker conditions limsup k→∞ ρB(k)<D∗ , limsup k→∞ ρE(k)<D∗ , where ρis the spectral radius of a matrix? TheadmissiblevaluesoftheparameterγusedintheproofofTheoremareγ>.Setting γ= and tracing all steps of the proof, we conclude that the assertion of exponential stability is not true. It can be seen, e.g.,from(), that, in such a case, only the stability is proved since, for x()m=δ(ε)andk≥, from ()weget x(k)≤ϕ(H)x()mγ–k/ =ϕ(H)x()m=ϕ(H)δ(ε)=ϕ(H)ε/ϕ(H)=ε. Let Kbe a set ofconsecutiveintegers such that k≥rfor every k∈K(ris an integer).In [] an exponential estimate of solutions of linearly perturbed linear systems y(k+)=A(k)+B(k)y(k), k∈K,() is studied assuming that matrices A(k)andB(k)aredefinedonK,B(k)≤β,k∈K, ·is a norm, and all solutions of the non-perturbed linear system x(k+)=A(k)x(k), k∈K,() satisfy the estimate x(k)≤qx(r)αk–r,k∈K,() for some positive constants qand α. Diblík et al. Advances in Difference Equations (2016) 2016:2 Page 19 of 20 Then, by Theorem , part (i) in [] any (forward) solution y(k)of()satisfies y(k)≤qy(r)(α+βq)k–r=q (α+βq)ry(r)(α+βq)k/,k∈K.() Tocompareourresultswiththosedescribedaboveweassumeα<in()(inthecaseof constantmatrixA,thisinequalityisequivalenttotheassumptionρ(A)<inTheorem). Compare estimate () with estimation ()inTheorem. The first one gives a qualitative result (since an estimate for qis not given and, therefore, it is not clear how small βshould be to guarantee the inequality α+βq<  and we cannot use ()fornumerical computations). The constants ϕ(H)and– ξ λmax(H) in () play the roles of the constants q(α+βq)–rand (α+βq)in (). We conclude that inequality () gives, together with qualitative information, a quantitative result as well and can be used for computational purposes. LetB(k)=forevery k=,,...,whereisan×nnullmatrix.Thenwecanset D:= and ξ=λmin(C). In such a case, Theorem becomes the following theorem. Theorem Let ρ(A)<.Let C be a fixed positive definite matrix,and let matrix H solve matrix equation (). Then system () is exponentially stable and,for any of its solutions x=x(k),the inequality x(k)≤– λmin(C) λmax(H)k/ϕ(H)x(),k≥, () holds where <–λmin(C) λmax(H)<. Comparinginequality ()with()(andassuming,forsimplicity,thatA(k)in()isa constantmatrix)wecanexplainthedifferencesbetweenbothinequalitiesinawaysimilar to the one we used above. Let us discuss the relationship between Theorem and Theorem .Itisobviousthat they are independent since, in the case of system (), we assume m≥. Moreover, setting D=and E(k)≡in system (), where is the zero matrix, in order to turn it into system (), does not turn the conclusion of Theorem to the conclusion of Theorem . A direct comparison leads to a conclusion that the latter is sharper. The reason for this is that the auxiliary inequalities (Lemmas , ,and), used in the proof of Theorem ,are rather imprecise and cannot imply the same result. Theauthorshave recentlypublishedpapers[,]onasimilar topicwherefurther related results can be found, including their application to scalar equations and their comparison with known results. In [] the boundedness character of positive solutions is studied. Let us refer as well to [] where some exact stability results are derived. Although our results cannot give such exact results, the advantage of our approach is that the estimates of the norms of the solutions are given by inequalities. Such estimates are not derived in []. Diblík et al. Advances in Difference Equations (2016) 2016:2 Page 20 of 20 Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details 1Brno University of Technology, Brno, Czech Republic. 2Kiev State University, Kiev, Ukraine. Acknowledgements The first author has been supported by the project No. LO1408, AdMaS UP - Advanced Materials, Structures and Technologies (supported by Ministry of Education, Youth and Sports of the Czech Republic under the National Sustainability Programme I). The second author has been supported by the project RP9051800400 (3.5 Support of International Mobility of Academic Staff, Faculty of Civil Engineering, Brno University of Technology) as a part of developing programme guaranteed by Ministry of Education, Youth and Sports of the Czech Republic). The third author has been supported by the Grant FEKT-S-14-2200 of Faculty of Electrical Engineering and Communication, Brno University of Technology. Received: 30 September 2015 Accepted: 23 December 2015 References 1. Agarwal, RP: Difference Equations and Inequalities: Theory, Methods and Applications, 2nd edn. Monographs and Textbooks in Pure and Applied Mathematics. Dekker, New York (2000) 2. 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